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Theorem isosolem 5490
Description: Lemma for isoso 5491. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isosolem (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))

Proof of Theorem isosolem
Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 5488 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
2 df-3an 898 . . . . . . 7 ((𝑎𝐴𝑏𝐴𝑐𝐴) ↔ ((𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴))
3 isof1o 5474 . . . . . . . . . . 11 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)
4 f1of 5153 . . . . . . . . . . 11 (𝐻:𝐴1-1-onto𝐵𝐻:𝐴𝐵)
5 ffvelrn 5327 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑎𝐴) → (𝐻𝑎) ∈ 𝐵)
65ex 112 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑎𝐴 → (𝐻𝑎) ∈ 𝐵))
7 ffvelrn 5327 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑏𝐴) → (𝐻𝑏) ∈ 𝐵)
87ex 112 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑏𝐴 → (𝐻𝑏) ∈ 𝐵))
9 ffvelrn 5327 . . . . . . . . . . . . 13 ((𝐻:𝐴𝐵𝑐𝐴) → (𝐻𝑐) ∈ 𝐵)
109ex 112 . . . . . . . . . . . 12 (𝐻:𝐴𝐵 → (𝑐𝐴 → (𝐻𝑐) ∈ 𝐵))
116, 8, 103anim123d 1225 . . . . . . . . . . 11 (𝐻:𝐴𝐵 → ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵)))
123, 4, 113syl 17 . . . . . . . . . 10 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑎𝐴𝑏𝐴𝑐𝐴) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵)))
1312imp 119 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵))
14 breq1 3794 . . . . . . . . . . 11 (𝑥 = (𝐻𝑎) → (𝑥𝑆𝑦 ↔ (𝐻𝑎)𝑆𝑦))
15 breq1 3794 . . . . . . . . . . . 12 (𝑥 = (𝐻𝑎) → (𝑥𝑆𝑧 ↔ (𝐻𝑎)𝑆𝑧))
1615orbi1d 715 . . . . . . . . . . 11 (𝑥 = (𝐻𝑎) → ((𝑥𝑆𝑧𝑧𝑆𝑦) ↔ ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦)))
1714, 16imbi12d 227 . . . . . . . . . 10 (𝑥 = (𝐻𝑎) → ((𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) ↔ ((𝐻𝑎)𝑆𝑦 → ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦))))
18 breq2 3795 . . . . . . . . . . 11 (𝑦 = (𝐻𝑏) → ((𝐻𝑎)𝑆𝑦 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
19 breq2 3795 . . . . . . . . . . . 12 (𝑦 = (𝐻𝑏) → (𝑧𝑆𝑦𝑧𝑆(𝐻𝑏)))
2019orbi2d 714 . . . . . . . . . . 11 (𝑦 = (𝐻𝑏) → (((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦) ↔ ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏))))
2118, 20imbi12d 227 . . . . . . . . . 10 (𝑦 = (𝐻𝑏) → (((𝐻𝑎)𝑆𝑦 → ((𝐻𝑎)𝑆𝑧𝑧𝑆𝑦)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏)))))
22 breq2 3795 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑐) → ((𝐻𝑎)𝑆𝑧 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
23 breq1 3794 . . . . . . . . . . . 12 (𝑧 = (𝐻𝑐) → (𝑧𝑆(𝐻𝑏) ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
2422, 23orbi12d 717 . . . . . . . . . . 11 (𝑧 = (𝐻𝑐) → (((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏))))
2524imbi2d 223 . . . . . . . . . 10 (𝑧 = (𝐻𝑐) → (((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆𝑧𝑧𝑆(𝐻𝑏))) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
2617, 21, 25rspc3v 2687 . . . . . . . . 9 (((𝐻𝑎) ∈ 𝐵 ∧ (𝐻𝑏) ∈ 𝐵 ∧ (𝐻𝑐) ∈ 𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
2713, 26syl 14 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
28 isorel 5475 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
29283adantr3 1076 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑅𝑏 ↔ (𝐻𝑎)𝑆(𝐻𝑏)))
30 isorel 5475 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑐𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
31303adantr2 1075 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑎𝑅𝑐 ↔ (𝐻𝑎)𝑆(𝐻𝑐)))
32 isorel 5475 . . . . . . . . . . . 12 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑐𝐴𝑏𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3332ancom2s 508 . . . . . . . . . . 11 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑏𝐴𝑐𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
34333adantr1 1074 . . . . . . . . . 10 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (𝑐𝑅𝑏 ↔ (𝐻𝑐)𝑆(𝐻𝑏)))
3531, 34orbi12d 717 . . . . . . . . 9 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝑎𝑅𝑐𝑐𝑅𝑏) ↔ ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏))))
3629, 35imbi12d 227 . . . . . . . 8 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → ((𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)) ↔ ((𝐻𝑎)𝑆(𝐻𝑏) → ((𝐻𝑎)𝑆(𝐻𝑐) ∨ (𝐻𝑐)𝑆(𝐻𝑏)))))
3727, 36sylibrd 162 . . . . . . 7 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
382, 37sylan2br 276 . . . . . 6 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ((𝑎𝐴𝑏𝐴) ∧ 𝑐𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
3938anassrs 386 . . . . 5 (((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) ∧ 𝑐𝐴) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4039ralrimdva 2416 . . . 4 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝑎𝐴𝑏𝐴)) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ∀𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4140ralrimdvva 2421 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦)) → ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
421, 41anim12d 322 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑆 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))) → (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏)))))
43 df-iso 4061 . 2 (𝑆 Or 𝐵 ↔ (𝑆 Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑆𝑦 → (𝑥𝑆𝑧𝑧𝑆𝑦))))
44 df-iso 4061 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑎𝐴𝑏𝐴𝑐𝐴 (𝑎𝑅𝑏 → (𝑎𝑅𝑐𝑐𝑅𝑏))))
4542, 43, 443imtr4g 198 1 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wo 639  w3a 896   = wceq 1259  wcel 1409  wral 2323   class class class wbr 3791   Po wpo 4058   Or wor 4059  wf 4925  1-1-ontowf1o 4928  cfv 4929   Isom wiso 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-po 4060  df-iso 4061  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-f1o 4936  df-fv 4937  df-isom 4938
This theorem is referenced by:  isoso  5491
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